Compton's Scattering Law
In 1923, Arthur Holly Compton observed that X-ray and gamma ray photons frequently scatter and lose energy (and gain wavelength) when they interact with electrons in matter. This phenomenon—which demonstrates that light has particle, as well as wave properties—has come to be known as Compton scattering. Observations have shown that this phenomenon can be characterized by the following Compton scattering equation (also known as Compton's law):
            λ      ′        -    λ    =            h                        m          e                ⁢        c              ⁢          (              1        -                  cos          ⁢                                          ⁢          ψ                    )      where λ and λ′ are the wavelengths, respectively, of the photon before and after the scattering; h is Planck's constant, me is the mass of the electron, c is the speed of light, and ψ is the angle by which the photon's heading changes, also known at the Compton scatter angle.
Compton Camera Principles
It is possible to create a device, known as a Compton camera, with a first detector and a second detector, each of which contains one or more detector elements, to cause and record incidents of Compton scattering, and from the detected information reconstruct a radioactive distribution from which the detected gamma and x-ray photons originated. In a Compton camera, the first detector, sometimes referred to as a scatter detector, has one or more first detector elements operable to scatter a photon interacting with a first detector element and to approximately measure an amount of energy lost by said photon as a result of said interaction. The second detector, in some Compton camera embodiments referred to as an absorption detector (although a Compton camera need not have second detector elements that fully absorb the detected photons), has multiple second detector elements operable to detect the scattered photon.
A Compton camera is typically associated with an instrument that is operable to record incidents in which a photon interacts with first and second detector elements, and in a manner that preserves information about the identities or positions of the first and second detector elements with which the photon interacted, and that also preserves information approximately indicating an amount of energy lost by the photon when it interacted with the first detector element. This can be done by partitioning the measured incidents into measurement bins. Typically, for each pair of first and second detector elements, Ne corresponding measurement bins are provided, each of which represents different detected energy levels. Each measurement bin could be tagged with three variables j, l, and k representing photons counted in the kth energy bin that interacted with the jth first element and the lth second element.
Because it is known that the energy of a photon is defined by the following equation:
  E  =      hc    λ  if one knows the initial wavelength λ of the detected gamma or x-ray photon (which can be known by knowing the radioactive isotope producing the radiation), then one can compute the post-scatter wavelength λ′ of the photon from the measured energy loss. From this information, one can deduce the approximate angle of the scatter in accordance with the Compton scattering equation. Thus, assuming that the initial wavelength λ of the detected gamma or x-ray photon is known, then each measurement bin would represent a count of detected photons with an approximate corresponding scatter angle ψ.
The inventor's article Reconstruction methods and completeness conditions for two Compton data models in the March 2005 edition of the Journal of the Optical Society of America, discusses the limitations of three prior art reconstruction methods and suggests two new reconstruction methods for Compton data. That article did not, however, set forth a methodological approach to selecting Compton camera shapes, configurations, positions, orientations, trajectory paths, and detector element sets to collect data for analysis using the two new reconstruction methods. Indeed, page 455 of the article stated that “[i]t is not immediately obvious what shapes, configurations, and motions of the detectors will satisfy [the] completeness conditions” described in that paper. The article also suggested that not until “an advantageous shape, configuration and motion of the detectors has been selected,” would it “be wise to build a full-scale Compton imaging system.”
Traditional paradigms for designing Compton cameras have been based on prior art reconstruction methods. But those paradigms are not optimal if the two Compton data models described in the 2005 paper are used to reconstruct.